Abstract

Shading is important for estimation of three-dimensional shape from the two-dimensional image, for instance, for distinguishing between the smooth occluding contour generated by the edge of a sphere and the sharp occluding contour generated by the edge of a disk. In order to use shading information to solve such problems, one must know the illuminant direction <b>L</b>. This is because variations in image intensity (shading) are caused by changes in surface orientation relative to the illuminant. Each illuminant direction <b>L</b> has a unique effect on the distribution of changes in image intensity d<i>I</i>, potentially potentially permitting the estimation of <i>L</i>. However, because d<i>I</i> is a function of <i>both</i><i>L</i> and the surface curvature, <i>L</i> can be estimated from the image only by making an assumption about the imaged surface curvature. One assumption that is sufficient to disentangle <i>L</i> and surface curvature is that changes in surface orientation are isotropically distributed. This condition is true of images of convex objects bounded by a smooth occluding contour and is true on average over all scenes. Estimates made by using this assumption agree with estimates of illuminant direction given by human subjects for images of natural objects, even when both are objectively wrong. Further, there is a significant correlation between the variance of these estimates and the variance of the human subjects’ estimates.

© 1982 Optical Society of America

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  1. B. K. P. Horn, "Understanding image intensities," Artif. Intell. 21, 201–231 (1977).
  2. A. P. Pentland, "Estimating illuminant direction to obtain shape from shading," in Digest of Topical Meeting on Advances in Vision (Optical Society of America, Washington, D.C., 1980).
  3. The change from a bandpass sensitivity for static images to a sensitivity that is a linear function of the rate of change in dynamic images may come about through summing of several bandpass channels, or it may be a change in the channels' characteristics; the specific manner in which this change comes about is irrelevant to the task at hand. How this change in sensitivity happens is a question of the algorithm the visual system is employing and not a question of what the system is doing. What is of interest to us now is that under normal viewing conditions the human sensitivity to changes in image intensity is approximately a linear function of the rate of change and thus can be modeled by the first derivative of image intensity.
  4. T. Cornsweet, Visual Perception (Academic, New York, 1970).
  5. E. L. Kirnov, Spectral Properties of Natural Fountains, G. Belkov, translator, NRC of Canada Technical Translation 439 (National Research Council, Ottawa, Canada, 1971).
  6. Here I refers to the image intensity at a point in the image, so that dI refers to the first derivative of image intensity. The context should make clear where, and in what direction, this derivative is taken.
  7. That is, calculate dI by using a particular image orientation at each point in each of the images, and then calculate the mean of those dI for each image. Thus one number is calculated for each image and image orientation.
  8. K. Stevens, "Surface perception from local analysis of texture and contour," Ph.D. Thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1979).
  9. This R is different from either the reflectance map1 or the bidirectional reflectance function, which describes how light is reflected for a particular L and V. The reflectance function defined here is a function of L, V, and N that, for particular values of L and V, generates the bidirectional reflectance function.
  10. A. Pentland, "The visual inference of shape: computation from local features," Ph.D. Thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1982).
  11. Image irradiance is what is actually being discussed here, not image intensity, which is the measured image irradiance. I will continue to refer to image intensity, however, as that is customary in the psychological literature. The distinction is not important here, as the two can simply be assumed to be numerically equal.
  12. This may be proved by noting that the surface normals on such an object are perpendicular to V at the boundary of such an object, and thus (given that the object is strictly convex) we may form a 1–1 onto map between the surface normals of the object and the Gaussian sphere, which satisfies Eq. (3).
  13. In that regions of the image are the fundamental unit of estimation, this theory bears a resemblance to gestalt theories. This resemblance is heightened by the similarity of the gestalt concept of pragnanz, as used to provide constraint on the interpretation of an image, to the assumption of zero mean change in surface orientation, which was used to provide the constraint necessary to estimate the illuminant direction.
  14. D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. Lond. Ser. B 207, 187–217 (1980).
  15. S. Ullman, The Interpretation Of Visual Motion (MIT Press, Cambridge, Mass., 1979).

1980

D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. Lond. Ser. B 207, 187–217 (1980).

1977

B. K. P. Horn, "Understanding image intensities," Artif. Intell. 21, 201–231 (1977).

Cornsweet, T.

T. Cornsweet, Visual Perception (Academic, New York, 1970).

Hildreth, E.

D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. Lond. Ser. B 207, 187–217 (1980).

Horn, B. K. P.

B. K. P. Horn, "Understanding image intensities," Artif. Intell. 21, 201–231 (1977).

Kirnov, E. L.

E. L. Kirnov, Spectral Properties of Natural Fountains, G. Belkov, translator, NRC of Canada Technical Translation 439 (National Research Council, Ottawa, Canada, 1971).

Marr, D.

D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. Lond. Ser. B 207, 187–217 (1980).

Pentland, A.

A. Pentland, "The visual inference of shape: computation from local features," Ph.D. Thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1982).

Pentland, A. P.

A. P. Pentland, "Estimating illuminant direction to obtain shape from shading," in Digest of Topical Meeting on Advances in Vision (Optical Society of America, Washington, D.C., 1980).

Stevens, K.

K. Stevens, "Surface perception from local analysis of texture and contour," Ph.D. Thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1979).

Ullman, S.

S. Ullman, The Interpretation Of Visual Motion (MIT Press, Cambridge, Mass., 1979).

Artif. Intell.

B. K. P. Horn, "Understanding image intensities," Artif. Intell. 21, 201–231 (1977).

Proc. R. Soc. Lond. Ser. B

D. Marr and E. Hildreth, "Theory of edge detection," Proc. R. Soc. Lond. Ser. B 207, 187–217 (1980).

Other

S. Ullman, The Interpretation Of Visual Motion (MIT Press, Cambridge, Mass., 1979).

A. P. Pentland, "Estimating illuminant direction to obtain shape from shading," in Digest of Topical Meeting on Advances in Vision (Optical Society of America, Washington, D.C., 1980).

The change from a bandpass sensitivity for static images to a sensitivity that is a linear function of the rate of change in dynamic images may come about through summing of several bandpass channels, or it may be a change in the channels' characteristics; the specific manner in which this change comes about is irrelevant to the task at hand. How this change in sensitivity happens is a question of the algorithm the visual system is employing and not a question of what the system is doing. What is of interest to us now is that under normal viewing conditions the human sensitivity to changes in image intensity is approximately a linear function of the rate of change and thus can be modeled by the first derivative of image intensity.

T. Cornsweet, Visual Perception (Academic, New York, 1970).

E. L. Kirnov, Spectral Properties of Natural Fountains, G. Belkov, translator, NRC of Canada Technical Translation 439 (National Research Council, Ottawa, Canada, 1971).

Here I refers to the image intensity at a point in the image, so that dI refers to the first derivative of image intensity. The context should make clear where, and in what direction, this derivative is taken.

That is, calculate dI by using a particular image orientation at each point in each of the images, and then calculate the mean of those dI for each image. Thus one number is calculated for each image and image orientation.

K. Stevens, "Surface perception from local analysis of texture and contour," Ph.D. Thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1979).

This R is different from either the reflectance map1 or the bidirectional reflectance function, which describes how light is reflected for a particular L and V. The reflectance function defined here is a function of L, V, and N that, for particular values of L and V, generates the bidirectional reflectance function.

A. Pentland, "The visual inference of shape: computation from local features," Ph.D. Thesis (Massachusetts Institute of Technology, Cambridge, Mass., 1982).

Image irradiance is what is actually being discussed here, not image intensity, which is the measured image irradiance. I will continue to refer to image intensity, however, as that is customary in the psychological literature. The distinction is not important here, as the two can simply be assumed to be numerically equal.

This may be proved by noting that the surface normals on such an object are perpendicular to V at the boundary of such an object, and thus (given that the object is strictly convex) we may form a 1–1 onto map between the surface normals of the object and the Gaussian sphere, which satisfies Eq. (3).

In that regions of the image are the fundamental unit of estimation, this theory bears a resemblance to gestalt theories. This resemblance is heightened by the similarity of the gestalt concept of pragnanz, as used to provide constraint on the interpretation of an image, to the assumption of zero mean change in surface orientation, which was used to provide the constraint necessary to estimate the illuminant direction.

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